Partial sums of Taylor series on a circle

Katsoprinakis, E. S., and Nestoridis, V. N.. "Partial sums of Taylor series on a circle." Annales de l'institut Fourier 39.3 (1989): 715-736. <http://eudml.org/doc/74848>.

@article{Katsoprinakis1989,
abstract = {We characterize the power series $\sum ^\{\infty \}_\{n=0\}c_ nz^n$ with the geometric property that, for sufficiently many points $z$, $\vert z\vert =1$, a circle $C(z)$ contains infinitely many partial sums. We show that $\sum ^\{\infty \}_\{n=0\}c_ nz^n$ is a rational function of special type; more precisely, there are $t\in \{\Bbb R\}$ and $n_ 0$, such that, the sequence $c_ne^\{\{\rm int\}\}$, $n\ge n_0$, is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results of [Katsoprinakis, Arkiv for Matematik]. We are led to consider special families of circles $C(z)$ with center $g(z)$ and investigate the possibility for a polynomial $R(z)$ to satisfy $R(z)\in C(z)$ for infinitely many $z$, $\vert z\vert =1$. These polynomials are related to the partial sums of the Taylor expansion of the center function $g(z)$. We also give necessary and sufficient conditions for the existence of infinitely many such polynomials $R(z)$.},
author = {Katsoprinakis, E. S., Nestoridis, V. N.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {715-736},
publisher = {Association des Annales de l'Institut Fourier},
title = {Partial sums of Taylor series on a circle},
url = {http://eudml.org/doc/74848},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Katsoprinakis, E. S.
AU - Nestoridis, V. N.
TI - Partial sums of Taylor series on a circle
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 3
SP - 715
EP - 736
AB - We characterize the power series $\sum ^{\infty }_{n=0}c_ nz^n$ with the geometric property that, for sufficiently many points $z$, $\vert z\vert =1$, a circle $C(z)$ contains infinitely many partial sums. We show that $\sum ^{\infty }_{n=0}c_ nz^n$ is a rational function of special type; more precisely, there are $t\in {\Bbb R}$ and $n_ 0$, such that, the sequence $c_ne^{{\rm int}}$, $n\ge n_0$, is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results of [Katsoprinakis, Arkiv for Matematik]. We are led to consider special families of circles $C(z)$ with center $g(z)$ and investigate the possibility for a polynomial $R(z)$ to satisfy $R(z)\in C(z)$ for infinitely many $z$, $\vert z\vert =1$. These polynomials are related to the partial sums of the Taylor expansion of the center function $g(z)$. We also give necessary and sufficient conditions for the existence of infinitely many such polynomials $R(z)$.
LA - eng
UR - http://eudml.org/doc/74848
ER -